Math Analysis Inverse Functions

In this lesson, our instructor Vincent Selhorst-Jones explains Inverse Functions through picture analogies which show how to denote the inverse and what comes out as a result. Youll study the requirements for reversing, how to do a horizontal line test, and the definition of an inverse function. Finding the inverse is easy with Vincents clear explanations about the different methods involved such as interchanging x and y or replacing y with the inverse. Before the four practice examples, youll also learn how to check inverses.

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Please, help me with this question:Find Laplace Transform of the following signal:x2 (t)=(1-(1-t)e-3t)u(t)

1 answer

Last reply by: John LinsSat Feb 11, 2017 8:06 PM

Post by John Linson February 11 at 06:25:28 PM

Hello professor vincent. Could you please help me to find this Laplace Transform?x2 (t)=(1-(1-t)e-3t)u(t)

1 answer

Last reply by: Professor Selhorst-JonesFri Mar 25, 2016 5:24 PM

Post by Ru Chigobaon November 18, 2015

Hi I need help with this problem :

Find the inverse of each relation or function, and then determine if the inverse is a function.1 f={1,3), (1,-1), (1,-3), (1,1)} f-1=

1 answer

Last reply by: Professor Selhorst-JonesTue Mar 17, 2015 11:24 PM

Post by thelma clarkeon March 17, 2015

is there no easier way to solve this it seem very confusing

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Last reply by: Professor Selhorst-JonesMon Oct 20, 2014 11:27 AM

Post by Saadman Elmanon October 18, 2014

It was a great clarification. Thanks,However, Inverse function is not only has to do with horizontal line test passing but also has to do with vertical test passing. You only stressed on Horizontal test passing. My professor stressed both.

1 answer

Last reply by: Professor Selhorst-JonesMon Jun 16, 2014 9:18 PM

Post by Joshua Jacobon June 15, 2014

Sorry if this is slightly vague but I'm a little but confused on the last example. Could you explain it in other words please?

Inverse Functions

A function does a transformation on an input. But what if there was some way to reverse that transformation? This is the idea of an inverse function: a way to reverse a transformation and get back to our original input.

To help us understand this idea, imagine a factory where if you give them a pile of parts, they'll make you a car. Now imagine another factory just down the road, where if you give them that car, they'll give you back the original pile of parts you started with. There is one process, but there is also an inverse process that gets you back to where you started. If you follow one process with the other, nothing happens.

It's important to note that not all functions have inverses. Some types of transformation cannot be undone. If the information about what we started with is permanently destroyed by the transformation, it cannot be reversed.

A function has an inverse if the function is one-to-one: for any a, b in the domain of f where a ≠ b, then f(a) ≠ f(b). Different inputs produce different outputs.

We can see this property in the graph of a function with the Horizontal Line Test. If a function is one-to-one, it is impossible to draw a horizontal line somewhere such that it will intersect the graph twice (or more).

Given some function f that is one-to-one, there exists an inverse function, f−1, such that for all x in the domain of f,

f−1

⎛⎝

f(x)

⎞⎠

= x.

In other words, when f−1 operates on the output of f, it gives the original input that went into f. [Caution: f−1 means the inverse of f, not [1/f]. In general, f−1 ≠ [1/f].]

We can figure out the domain and range for f−1 by looking at f. Since the set of all outputs is the range of f, and f−1 can take any output of f, the domain of f−1 is the range of f. Likewise, f−1 can output all possible inputs for f, so the range of f−1 is the domain of f.

The inverse of f−1 is simply f. This makes intuitive sense: if you do the opposite of an opposite, you end up doing the original thing.

Visually, f−1 is the mirror of f over y=x. This is because f−1 swaps the outputs and inputs from f, which is the same thing as swapping x and y by mirroring over y=x.

To find the inverse to a function, we effectively need a way to "reverse" the function. This can be a little bit confusing at first, so here is a step-by-step guide for finding inverse functions.

While this method will produce the inverse if followed correctly, it is not perfect. Notice that in steps #2 and #3 above, the equations are completely different, yet they still use the same x and y. Technically, it is not possible for x and y to fulfill both of these equations at the same time. What's really happening is that when we swap in #3, we're actually creating a new, different y. The first one stands in for f(x), but the second stands in for f−1(x). This implicit difference between y's can be confusing, so be careful. Make a note on your paper where you swap x and y so you can see the switch to "inverse world".

Taking inverses can be difficult: it's easy to make a mistake. This means it's important to check your work. By definition, f−1 ( f(x) ) = x. This means if you know what f−1(x) and f(x) are, you can just compose them! If it really is the inverse, you'll get x. Furthermore, since we know f( f−1 (x) ) = x as well, you can compose them in either order when checking.

Inverse Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.